Can this integral be done using elementary techniques?

[Bipolarity]

Homework Statement

$\displaystyle \int \frac{dx}{(x^{2}+a^{2})^{3/2}}$

Homework Equations

The Attempt at a Solution

A substitution does not seem to work. I know that a closed form solution exists however, just curious if it can be done by the standard techniques usually taught in calculus. Thanks!

BiP

 

[Infrared]

The substitution x=asinh\theta seems to work.

Edit: x=atan\theta also works if you don't mind integrating the secant function.

 

[Goa'uld]

...
Edit: x=atan\theta also works if you don't mind integrating the secant function.

Actually, you need to integrate cosine after that substitution. It may be the easiest way.

 

[Infrared]

Actually, the integrand simplifies to (cosθ)/a^2 after that substitution. It may be the easiest way.

You are right of course. Somehow I turned \frac{1}{sec\theta} into \frac{1}{cos\theta}

 

[Mark44]

The presence of x² + a² in the denominator makes this integral a candidate for trig substitution. Same goes for the difference of squares, either x² - a² or a² - x².

What I do in these situations is draw a right triangle, and label the sides and hypotenuse according to whether I'm dealing with a sum of squares or a difference. If it's a sum of squares, as in this problem, I label the vertical side as x and the horizontal side as a. This means that the hypotenuse is √(x² + a²). If θ is the angle across from x, then tanθ = x/a, so sec²θdθ = dx/a, or dx = asec²θdθ. After this, replace x and dx in the integral with θ and dθ and integrate.

A similar analysis can be done for either of the two differences of squares.